TR00-059 Authors: Omer Reingold, Ronen Shaltiel, Avi Wigderson

Publication: 14th August 2000 08:40

Downloads: 1819

Keywords:

On an input probability distribution with some (min-)entropy

an {\em extractor} outputs a distribution with a (near) maximum

entropy rate (namely the uniform distribution).

A natural weakening of this concept is a condenser, whose

output distribution has a higher entropy rate than the

input distribution (without losing

much of the initial entropy).

In this paper we construct efficient explicit condensers.

The condenser constructions combine

(variants or more efficient versions of)

ideas from several works,

including the block extraction scheme of Nisan and Zuckerman,

the observation made by Nisan and Ta-Shma

that a failure of the block extraction scheme is also useful,

the recursive ``win-win'' case analysis of Impagliazzo Shaltiel and

Wigderson, and the error correction of random sources used by Trevisan.

As a natural byproduct, (via repeated iterating of condensers),

we obtain new extractor constructions.

The new extractors

give significant qualitative improvements over

previous ones for sources of arbitrary min-entropy; they are nearly

optimal simultaneously in the main two parameters - seed length and

output length. Specifically, our extractors can make any of these

two parameters optimal (up to a

constant factor), only at a poly-logarithmic loss in the other.

Previous constructions require polynomial loss in both cases for

general sources.

We also give a simple reduction converting ``standard'' extractors

(which are good for an average seed) to

``strong'' ones (which are good for most seeds), with essentially the

same parameters.

With it, all the above improvements apply to strong extractors as well.